If A and B are two events such that `P(B) = 2/5 and P(Acap B) = 1/5," then : " P(A cup B')=`

Options

A. `1/5`
B. `2/5`
C. `4/5`
D. `3/5`

Answer

To solve the problem, we need to find \( P(A \cup B') \) given that \( P(B) = \frac{2}{5} \) and \( P(A \cap B) = \frac{1}{5} \).

### Step-by-Step Solution:

1. **Understanding the Events**:
- We have two events, A and B.
- \( P(B) = \frac{2}{5} \) means the probability of event B occurring is \( \frac{2}{5} \).
- \( P(A \cap B) = \frac{1}{5} \) means the probability that both events A and B occur is \( \frac{1}{5} \).

2. **Finding \( P(B') \)**:
- The complement of event B, denoted as \( B' \), represents the event that B does not occur.
- The probability of the complement of an event is given by:
\[ P(B') = 1 - P(B) \]
- Substituting the value of \( P(B) \):
\[ P(B') = 1 - \frac{2}{5} = \frac{3}{5} \]

3. **Using the Formula for Union of Events**:
- We need to find \( P(A \cup B') \). The formula for the probability of the union of two events is:
\[ P(A \cup B') = P(A) + P(B') - P(A \cap B') \]
- We will need to find \( P(A) \) and \( P(A \cap B') \).

4. **Finding \( P(A) \)**:
- We can express \( P(A) \) using the formula:
\[ P(A) = P(A \cap B) + P(A \cap B') \]
- We know \( P(A \cap B) = \frac{1}{5} \), but we do not yet know \( P(A \cap B') \).

5. **Finding \( P(A \cap B') \)**:
- We can find \( P(A \cap B') \) using the relationship:
\[ P(A \cap B') = P(A) - P(A \cap B) \]
- However, we need \( P(A) \) first. To find \( P(A) \), we can use the total probability:
\[ P(A) = P(A \cap B) + P(A \cap B') = P(A \cap B) + P(A) - P(A \cap B) \]
- Rearranging gives us:
\[ P(A) = P(A \cap B) + P(A \cap B') \]

6. **Calculating \( P(A \cup B') \)**:
- Now substituting the known values:
\[ P(A \cup B') = P(A) + P(B') - P(A \cap B') \]
- We know \( P(B') = \frac{3}{5} \) and \( P(A \cap B) = \frac{1}{5} \).

7. **Final Calculation**:
- Since we do not have \( P(A) \) directly, we can express \( P(A \cup B') \) as:
\[ P(A \cup B') = P(A \cap B) + P(B') = \frac{1}{5} + \frac{3}{5} = \frac{4}{5} \]

### Final Answer:
\[ P(A \cup B') = \frac{4}{5} \]